Efficient Evaluation of Double Surface Integrals in Time-Domain Integral Equation Formulations

A new integration approach is presented for accurately calculating time-domain EFIE, MFIE, and CFIE matrix elements over triangular domains. It mainly consists of a radial integration scheme for handling weakly singular and near-hypersingular inner integrals and some new smoothing techniques for treating outer two-dimensional (2-D) integrals. The proposed approach has sufficient generality and efficiency for solving time-domain integral equations (TDIE) with arbitrary types of temporal basis functions and temporal discretization schemes, such as marching-on-in-time (MOT), marching-on-in-degree (MOD), and finite difference delay modeling/convolution quadrature (FDDM/CQ), etc. The numerical results for calculating some typical integrals are given to demonstrate its capability, with high accuracy and rapid convergence rate achieved.

I. Introduction

Time-Domain integral equation (TDIE)-based methods have prevalent usage for the study of transient electromagnetic problems, and various techniques and temporal basis functions have been used to solve the TDIEs so as to improve their stability, accuracy and efficiency [1]–[7]. However, differences in temporal basis functions and solution techniques may result in significant difference in the integral kernels of the double surface integrals, which largely increases the difficulties in fast and accurately evaluating the matrix elements of TDIE. Unfortunately, traditional singularity subtraction methods [8]–[12] are not always applicable to the TDIE. On the other hand, the methods of singularity cancellation [13]–[19] require the numerical evaluation of the four-dimensional (4-D) integrations.